We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.

翻译：我们证明,美元平面点之间的单位距离最多为1.94美元,比以前最好的约束值80美元有所改进,我们还为几个小值的美元提供了更好的上下限。我们还证明了一些跨越莱马的变种,并改进了一些不变因素。